I've been building computers for my friends and I for 25 years and the two worst "random stability" issues I had were high quality but aging PSUs.
I've had one case of Corsair memory which went faulty after a year (was replaced without question by the supplier) and around 3 PSU failures.
However, on the 3 times I've done upgrades (typically motherboard + RAM + CPU) in that time I've been able to keep my existing PSU without stability issues.
So I wouldn't say it's "insanity" to keep your current PSU when upgrading, but based on your experience if I had stability issues it may be the first thing I test.
No need to replace a quality PSU until you start having issues.